Alternating Current (AC) Circuits
AC Voltages and PhasorsResistors, Inductors and Capacitors in AC CircuitsRLC CircuitsPower and ResonanceTransformers
AC SourcesAC sources have voltages and currents that vary sinusoidally.The current in a circuit driven by an AC source also oscillates at the same frequency.The standard AC frequency in the US is 60 Hz (337 rad/s).
)sin()( max tVtv ωΔ=Δ
ωπ21
==f
T
t
ΔVmax
−ΔVmax
Δv(t)
PhasorsConsider a sinusoidal voltage source: )sin()( max tVtv ωΔ=Δ
This source can be represented graphically as a vector called a phasor:
ωtT
ΔVmaxΔv(t)b
c
b
c
d
e
d
aa e
Resistors in AC Circuits
R
R
vtVvv
Δ=Δ=Δ−Δωsin
0
max
tItRV
Rvi R
R ωω sinsin maxmax =
Δ=
Δ=
RVI max
maxΔ
=
Since both have the same arguments in the sine term, iR and ΔvR are in line, they are in phase.
tRIvR ωsinmax=Δ
tIiR ωsinmax=
tRIRit R ω22max
2 sin)( ==P
tRIRit R ω22max
2 sin)( ==P
maxmax
22max
707.02
21
III
RIRI
rms
rmsav
==
=⎟⎠⎞
⎜⎝⎛=P
maxmax 707.02
VVVrms Δ=Δ
=Δ
Root Mean Square (RMS) Current and Voltage
Inductors in AC Circuits
dtdiLtV
vv L
=Δ
=Δ−Δ
ωsin
0
max
tdtL
Vdi ωsinmaxΔ=
tL
VdiiL ωω
cosmaxΔ−== ∫
⎟⎠⎞
⎜⎝⎛ −
Δ=
2sinmax πω
ωt
LViL
LXV
LVI maxmax
maxΔ
=Δ
=ω
LX L ω=
tXIv LL ωsinmax−=Δ
Inductive Reactance
For a sinusoidal voltage, the current in the inductor always lags the voltage across it by 90°.
Capacitors in AC Circuits
tVvv C ωsinmaxΔ=Δ=Δ
tVCq ωsinmaxΔ=
⎟⎠⎞
⎜⎝⎛ +Δ=
Δ==
2sin
cos
max
max
πωω
ωω
tVCi
tVCdtdqi
C
C
CXVVCI max
maxmaxΔ
=Δ=ω
CXC ω
1= Capacitive
Reactance
tXIv CC ωsinmax=Δ
For a sinusoidal voltage, the current in the capacitor always leads the voltage across it by 90°.
RLC Series CircuittVv ωsinmaxΔ=Δ
( )φω −= tIi sinmax
tVtRIv RR ωω sinsinmax Δ==Δ
tVtXIv LLL ωπω cos2
sinmax Δ=⎟⎠⎞
⎜⎝⎛ +=Δ
tVtXIv CCC ωπω cos2
sinmax Δ−=⎟⎠⎞
⎜⎝⎛ −=Δ
CLR vvvv Δ+Δ+Δ=Δ
Using Phasors in a RLC Circuit( )
( ) ( )( )22
maxmax
2maxmax
2maxmax
22max
CL
CL
CLR
XXRIV
XIXIRIV
VVVV
−+=Δ
−+=Δ
Δ−Δ+Δ=Δ
( )22
maxmax
CL XXR
VI−+
Δ=
( )22CL XXRZ −+≡ Impedance
ZIV maxmax =Δ
⎟⎠⎞
⎜⎝⎛ −
= −
RXX CL1tanφ
R = 425 ΩL = 1.25 HC = 3.5 μFω
= 377 s-1
ΔVmax = 150 V
( )( ) Ω=== − 7585.3377
111 FsC
XC μω
( )( ) Ω=== − 47125.1377 1 HsLX L ω
( ) ( ) ( ) Ω=Ω−Ω+Ω=−+= 513758471425 2222CL XXRZ
AVZVI 292.0
513150max
max =Ω
=Δ
=
°−=⎟⎠⎞
⎜⎝⎛
ΩΩ−Ω
=⎟⎠⎞
⎜⎝⎛ −
= −− 34425
758471tantan 11
RXX CLφ
( )( ) VARIVR 124425292.0max =Ω==Δ
( )( ) VAXIV LL 138471292.0max =Ω==Δ
( )( ) VAXIV CC 221758292.0max =Ω==Δ
( ) tVvC 377cos221−=Δ
( ) tVvL 377cos138=Δ
( ) tVvR 377sin124=Δ
Example 33.4
Power in an AC Circuit( ) tVtIvi ωφω sinsin maxmax Δ−=Δ=P
( )φωωφω
ωφωφω
sincossincossin
sinsincoscossin
maxmax2
maxmax
maxmax
ttVItVI
tttVI
Δ−Δ=
−Δ=
P
P
φcos21
maxmax VIav Δ=P
φcosrmsrmsav VI Δ=P
RIVvR maxmax cos =Δ=Δ φ
22max
max
maxmax RIIV
RIVI rmsrmsav =Δ
⎟⎠⎞
⎜⎝⎛ Δ=P
RIrmsav2=P
For a purely resistive load, φ=0
rmsrmsav VI Δ=P
No power loss occurs in an ideal capacitor or inductor.
Resonance in Series RLC Circuits
ZVI rms
rmsΔ
=( )22
CL
rmsrms
XXR
VI−+
Δ=
A circuit is in resonance when its current is maximum.
Irms =max when XL -XC =0
CL
XX CL
00
1ω
ω =
=
LC1
0 =ω Resonance Frequency
Power in Series RLC Circuits ( ) ( )
( )22
2
2
22
CL
rmsrmsrmsav XXR
RVRZVRI
−+Δ
=Δ
==P
( ) ( )220
22
222 1 ωω
ωωω −=⎟
⎠⎞
⎜⎝⎛ −=−
LC
LXX CL
( )( )22
02222
22
ωωω
ω
−+
Δ=
LR
RVrmsavP
( )R
Vrmsav
2
max,Δ
=P ω=ω0
RL00 ω
ωω
=Δ
=Q Quality Factor
Transformersdt
dNV BΦ−=Δ 11
dtdNV BΦ
−=Δ 22
11
22 V
NNV Δ=Δ
2211 VIVI Δ=Δ
LRVI 2
2Δ
=eqRVI 1
1Δ
=
Leq RNNR
2
2
1⎟⎟⎠
⎞⎜⎜⎝
⎛=
Power TransmissionPower is transmitted in high voltage over long distances and then gradually stepped down to 240V by the time it gets to a house.
22 kV 230 kV 240 V
P
g = 20 MWCost = 10¢/kWh
AV
WV
I g 8710230
10203
6
=××
=Δ
=P
( ) ( ) kWARI 15287 22 =Ω==P
( )( )( ) 36$1.0$2415 == hkWTotalCost
For 22kV transmission:
Total Cost = $4100
For Next ClassMidterm 2 Review on MondayMidterm 2 on TuesdayReading Assignment for Wednesday
Chapter 34 – Electromagnetic Waves
WebAssign: Assignment 11 (due Monday, 5 PM)
